If there's a specific example from Wikipedia that you'd like to discuss in more depth, please do so.

Getting back to my example:

[tex](x+y)^2y' = 0[/tex]

The equation is solved by "two" equations:

[tex]y_1 = -x[/tex]

[tex]y_2 = C[/tex]

Suppose we set initial conditions to something very simple, like:

[tex]y(0) = 0[/tex]

Then there would be "two" curves (or lines, for precisely), which would satisfy the equation and initial conditions. That is, the line y=-x passes through the origin, as does the line y=0, both of which are solutions to the differential equation, and both of which satisfy the initial conditions.

So it would appear that failure of uniqueness is something that happens here.

My question is just, do we say that the whole system is singular, or just the y=x part, or what?